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Let W = (p : q : r) be a point. Kp(W) is the locus of pivots of all pK+ invariant under the isoconjugation with pole W. These cubics form the class CL014 of cubics. The locus of the common point Q of the three asymptotes is Kc(W), member of the class CL015.

Now, instead of fixing the pole W, we fix the pivot P = (u : v : w). The locus of poles of all pK+ with pivot P is Kw(P) and the locus of the common point Q of the three asymptotes is Kc'(P).

The cubics Kw(P) and Kc'(P) form the classes CL017 and CL018 respectively.

The general equation of Kw(P) is :

Kw(P) is also psK(P^2 x ccP, G, P^2) in Pseudo-Pivotal Cubics and Poristic Triangles, where ccP is the complement of the complement of P.

For example, Kw(X1) = K207 = psK(X2308, X2, X6) and Kw(X4) = K208 = psK(X5 x X393, X2, X53).

The barycentric quotient of Kw(P) by P is a cubic of the class CL063. For instance, K207 and K208 correspond to K702 and K071 respectively.

Points on Kw(P) and other properties

Kw(P) contains the following points :

• A, B, C.

The tangents at A, B, C to Kw(P) concur at the point u^2(v + w + 2u) : : , the barycentric product of W3 = P^2 and ccP = v + w + 2u : : , the homothetic of P under h(G,1/4) or equivalently the complement of the complement of P.

• the midpoints Ma, Mb, Mc of ABC.

• W0 = G/P (cevian quotient) = u(v + w - u) : : , the center of the circumconic with perspector P.

This gives the class CL009 of pK+ with asymptotes concurring at G.

• W1= (v + w)(v + w + 2u) : : , cevian quotient of G and ccP.

The cubic pK(W1, P) is always a pK++ (a central cubic) and it is the only non degenerate cubic of this type.

Its asymptotes concur at ccP.

• W2 = 1 / (v + w) : : , isotomic conjugate of the complement of P or cevapoint of G and P.

pK(W2, P) is in general not a pK++ and its asymptotes concur at Q2 with rather complicated coordinates.

• W3 = u^2 : : , barycentric square of P.

The cubic pK(W3, P) degenerates into the cevian lines of P.

The points G, W1, W2, W3 are collinear on the line Lp with equation (v^2 - w^2)x + cyclic = 0.

• W4 = u(v + w + 2u) : : . See CL049.

• W5 = u / (v^2 + w^2 + v w - u^2) : : .

• W6 = (v + w + 2u) / (v^2 + w^2 - u^2) : : .

• W7 on W2W4

• W8 = u^2(v + w + 2u)(v^2 + w^2 + vw - uv - uw) : : .

• W9 = u(v + w + 2u)(u^2 + vw + 2uv + 2uw) / (-u^2 + 2vw + uv + uw) : : .

• W10 = W2W6 /\ W4W5.

• W11 = W3W5 /\ W4W6.

• W12 = u^2(v + w)(v + w + 2u) / (v^2 + w^2 - u^2 + vw + uv + uw) : : .

• W13 = W3W6 /\ W5W9.

***

These points Wi (apart W3) are the poles of thirteen non degenerate pK+ with pivot P, most of them with reasonnably simple equations. The asymptotes concur at a corresponding point Qi. See these points in CL018.

There are actually infinitely many more points on Kw(P) giving infinitely many other pK+ with pivot P. Indeed, Kw(P) is invariant under two transformations f and g, one being the inverse of the other. If we denote by M* the isoconjugate of M under the isoconjugation that swaps P^2 and ccP, then f(M) = (G/M)* and g(M) = G/M* where / denotes Ceva conjugation.

The points Wi above are only special (and relatively simple) members of several infinite chains of points on Kw(P), for example :

... –> W11 –> W0 –> W4 –> W5 –> ...

... –> #W5 –> W10 –> W1 –> W3 –> W6 –> ...

... –> #W6 –> W13 –> W8 –> W2 –> W12 –> ...

where –> is the transformation f.

One jumps from one chain to another by tangential transformation i.e. by associating to one point Wi its tangential #Wi in Kw(P).

For example, W6 = #W0, W1 = #W4 and W8 = #W3.